Question

If $l^r\left(x\right)$ means $\log \log \log .........x,$ the $\log$ being repeated r times, then $\int {\left[xl\right(x\left)l^2\right(x\left)l^3\right(x)......l^r(x\left)\right]}^{-1}dx$ is equal to

• A. $l^{r+1}\left(x\right)+C$
• B. $\frac{l^{r+1}\left(x\right)}{r+1}+C$
• C. $l^r\left(x\right)+C$
• D. none of these

Answer 1

The correct option is A $l^{r+1}\left(x\right)+C$

Putting $l^{r+1}\left(x\right)=t$
$\Rightarrow \frac{1}{xl\left(x\right)l^2\left(x\right)...l^r\left(x\right)}dx=dt$
Consider $\int \frac{1}{xl\left(x\right)l^2\left(x\right)l^3\left(x\right)....l^r\left(x\right)}dx$
$=\int 1dt$
$=t+C$
$I=l^{r+1}\left(x\right)+C$